AIS - Index Theorems (2009)

Venue: IIT Bombay
Dates: 6 July - 1 Aug


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name M. S. Raghunathan  R. S. Kulkarni,  A. Ranjan  and A. R. Shastri
Mailing Address

msr at math.tifr.res.ini

punekulk at,  aranjan at,  ars at

Advanced Instructional Schoolon Atiyah-Singer Index Theorem is being organised in IIT, Bombay, Mumbai in July-August 2009. The school is sponsored by NBHM.

Speakers and Syllabus 


Ravi Kulkarni IIT, Bombay
Nitin Nitsure TIFR, Mumbai
M.S. Raghunathan TIFR, Mumbai
A. Ranjan IIT, Bombay
Harish Seshadri IISc, Bangalore
A. R. Shastri IIT, Bombay



Basic Differential Geometry. Manifolds, differential forms, Stokes theorem, tangent and cotangent bundles, Riemannian metrics, connections on vector bundles, jet bundles of vector bundles, linear differential operators between vector bundles, Peetre theorem, Laplacian, statement of Hodge theorem.

Overview of singular cohomology, CW complexes, vector bundles, characteristic classes (Chern and Euler) via axioms, existence of characteristic classes via classifying spaces, elementary facts about cobordism and Morse theory, Thom's theorem about spherical cohomology classes on manifolds.

Operations with vector bundles, pullbacks, homotopy invariance, clutching construction, collapsing a closed subspace, the relative K functors, long exact sequence for a pair, Bott periodicity, half-exact functors, isomorphism of rational K theory with cohomology with rational coefficients via Chern character, Thom isomorphism in (integral) K theory for complex vector bundles.

Twisted signature operator, Thom isomorphism for K tensored with rationals, comparison theorem for Thom isomorphism in rational K theory and rational cohomology, Dolbeault complex, Hodge Index Theorem, Hirzebruch-Riemann-Roch Theorem, Dirac operators.

Elliptic operators, a priori inequalities, finiteness theorem for kernel and cokernel, the heat equation, the heat kernel and its asymptotics. Proof of the Atiyah-Singer Index Theorem

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Selected Applicants

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